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A241139 Number of nonprimes in factorization of n! over distinct terms of A050376.

%I A241139 #21 Sep 17 2019 20:02:00
%S A241139 0,0,1,1,2,2,3,3,3,3,4,4,5,6,7,7,4,4,5,5,6,6,8,9,10,10,9,9,11,11,12,
%T A241139 12,10,9,8,8,9,10,11,11,12,12,11,12,14,14,16,15,15,15,13,13,14,14,14,
%U A241139 14,16,16,16,16,17,19,21,21,18,18,19,16,14,14,16,16,17
%N A241139 Number of nonprimes in factorization of n! over distinct terms of A050376.
%D A241139 V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].
%H A241139 Amiram Eldar, <a href="/A241139/b241139.txt">Table of n, a(n) for n = 2..1000</a>
%H A241139 S. Litsyn and V. S. Shevelev, <a href="http://www.emis.de/journals/INTEGERS/papers/h33/h33.Abstract.html">On factorization of integers with restrictions on the exponent</a>, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.
%F A241139 a(n) = A177329(n) - A055460(n).
%e A241139 Factorization of 4! over distinct terms of A050376 is 4! = 2*3*4. This factorization contains only one A050376-nonprime. So a(4)=1.
%t A241139 b[n_] := 2^(-1 + Position[Reverse@IntegerDigits[n, 2], _?(# == 1 &)]) // Flatten; a[n_] := Module[{np = PrimePi[n]}, v = Table[0, {np}]; Do[p = Prime[k]; Do[v[[k]] += IntegerExponent[j, p], {j, 2, n}], {k, 1, np}]; Length[Select[(b /@ v) // Flatten, # > 1 &]]]; Array[a, 73, 2]  (* _Amiram Eldar_, Sep 17 2019 *)
%o A241139 (PARI) a(n)={my(f=factor(n!)[,2]); sum(i=1, #f~, hammingweight(f[i]>>1))} \\ _Andrew Howroyd_, Sep 17 2019
%Y A241139 Cf. A177329, A177333, A177334, A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123, A241124.
%K A241139 nonn
%O A241139 2,5
%A A241139 _Vladimir Shevelev_, Apr 16 2014
%E A241139 More terms from _Peter J. C. Moses_, Apr 17 2014